Difference Equations and Reconstruction of Local Gopakumar-Vafa Spectra for Threefold Flops

RESUMO

We study difference equations for the local Gromov-Witten theory of a contractible rational curve in a Calabi-Yau threefold from the viewpoint of reconstruction. Starting from the Bryan-Katz-Leung multiple cover decomposition, we associate to all nonzero local GV degree sectors on an exceptional ray a single central difference forcing term governed by explicit Fejér-type Laurent kernels. Our main result is an information equivalence theorem: once the exceptional ray and its lattice normalization are fixed, the full quantum forcing, its regular specialization at $\lambda=0$, and the local genus zero Gopakumar-Vafa spectrum determine one another. In particular, every local GV multiplicity is recovered from the classical forcing by an explicit Möbius inversion. In the primitive normalization relevant to a simple flopping contraction, the same classical forcing also recovers the noncommutative and commutative widths of the contraction algebra: the former is the order at $q=1$ of the exponentiated forcing and the latter is its first $q$-coefficient. For finite support we further determine the minimal Laurent polynomial denominator clearing operator on the finest common shift lattice. Its cyclotomic factorization is strictly coarser than the GV spectrum. Length two, $E_6$, $E_7$, and $E_8$ examples exhibit the resulting hierarchy of information loss: Kollár length does not determine the forcing, the cyclotomic skeleton does not determine the GV support, and even the complete local difference data do not determine the analytic type of the flop.